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Geometry and Trigonometry / Circles Difficulty: Medium

The figure presents a circle in the x y plane, with the origin labeled O. The center of the circle lies at the origin. There are 4 points, T, R, Q, and P, labeled clockwise on the circle. Point T lies on the x axis, directly to the left of the origin, and has coordinates negative 1 comma 0. Point R lies above the x axis, and to the left of the y axis. Point Q lies above the x axis and to the right of the y axis. Point P lies on the x axis, directly to the right of the origin, and has coordinates 1 comma 0. Points R and Q appear to be symmetric with respect to the y axis. Four line segments are drawn from the origin to each of the four points on the circle.

In the xy-plane above, points P, Q, R, and T lie on the circle with center O. The degree measures of angles $P O Q$ and $R O T$ are each 30°. What is the radian measure of angle $Q O R$ ?

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Explanation

Choice C is correct. Because points T, O, and P all lie on the x-axis, they form a line. Since the angles on a line add up to $180 degrees$ , and it’s given that angles POQ and ROT each measure $30 degrees$ , it follows that the measure of angle QOR is $180 degrees minus 30 degrees, minus 30 degrees, equals 120 degrees$ . Since the arc of a complete circle is $360 degrees$ or $2 pi$ radians, a proportion can be set up to convert the measure of angle QOR from degrees to radians: $the fraction 360 degrees over 2 pi radians, equals, the fraction 120 degrees over x radians$ , where x is the radian measure of angle QOR. Multiplying each side of the proportion by $2 pi x$ gives $360 x equals 240 pi$ . Solving for x gives $the fraction 240 over 360 times pi$ , or $two thirds pi$ .

Choice A is incorrect and may result from subtracting only angle POQ from $180 degrees$ to get a value of $150 degrees$ and then finding the radian measure equivalent to that value. Choice B is incorrect and may result from a calculation error. Choice D is incorrect and may result from calculating the sum of the angle measures, in radians, of angles POQ and ROT.